Symmetric Airy beamsPablo Vaveliuk,1,2,* Alberto Lencina,2,3 Jose A. Rodrigo,4 and Oscar Martinez Matos4
1Faculdade de Tecnologia, Servicio Nacional de Aprendizagem Industrial SENAI-CimatecAv. Orlando Gomes 1845 41650-010, Salvador, Bahia, Brazil
2Centro de Investigaciones Ópticas (CICBA-CONICET), Cno. Parque Centenario s/n 1897 Gonnet,La Plata, Pcia. de Buenos Aires, Argentina
3Departamento de Física, Fac. de Cs. Exactas, Universidad Nacional de La Plata, c.c. 67,1900 La Plata, Pcia. de Buenos Aires, Argentina
4Departamento de Óptica, Facultad de Ciencias Físicas, Universidad Complutense de MadridAv. Complutense s/n 28040 Madrid, Spain*Corresponding author: [email protected]
Received December 27, 2013; revised March 12, 2014; accepted March 12, 2014;posted March 13, 2014 (Doc. ID 203843); published April 9, 2014
Since the demonstration of the so-called acceleratingAiry beam [1,2] in 2007, the research related to thesepeculiar self-bending beams has grown quickly, so muchin basic research [3] as in its generation, control, andapplications [4]. Nowadays, it constitutes an importantresearch area of optical beam propagation and design[5]. This beam is a separable solution of the Helmholtzequation in the paraxial regime for rectangular coordi-nates, and therefore, it can be generated in one or twotransverse dimensions. Its field profile is an exponen-tially modulated Airy function being, a square-integrableprofile encompassing a finite energy content. The Airyfunction has no defined symmetry in rectangular coordi-nates, but if extended to cylindrical ones, it results ina beam having circular symmetry with an Airy radialprofile, called abruptly autofocusing beam or circularAiry beam (CAB) [6–8]. These beams were experimen-tally demonstrated [9,10] and possess an unusual feature.As they propagate the profile is accelerated toward thecenter of symmetry yielding a parabolic caustic surfaceof revolution. This surface collapses on axis to an auto-focus with high intensity. This autofocus is a conse-quence of the optical field structure itself, and not byany nonlinear effect. The research on CABs has grownquickly in the last years, encouraged by the potentialapplications in laser medicine, laser ablation, and otherlinear or nonlinear optical settings [11].CABs are necessarily two-dimensional transverse
structures since they were conceived as radially symmet-ric beams in cylindrical coordinates. A main question forAiry-type structures concerns the possible link: circularsymmetry ↔ autofocusing properties. Indeed, theextension of the Airy beam design to the cylindricalframe compels the resulting beam to acquire a radialsymmetry. And this symmetry that allows the appearance
of autofocusing features. Nevertheless, the rectangularcoordinates represent the natural symmetry of aconventional Airy beam. Then, is it feasible an Airy-likebeam with autofocusing characteristics in a rectangularframe? We underline that the superposition of two Airybeams, having opposite accelerations and having theirprincipal lobes mostly overlapped, yields an autofocus-ing structure. Such a beam, called dual Airy beam(DAB), has been recently analyzed [12], although its au-tofocusing properties were not considered in that work.However, even if a manipulation of a DAB is theoreticallypossible, in practice, the experimental generation oftwo independent Airy beams overlapped in a fashionto obtain a DAB exhibiting autofocusing propertiesseems to be an extremely complex technical task.
In this Letter we develop a formalism to generate anAiry beam having rectangular symmetry and autofocus-ing properties that can be experimentally created aseasily as a conventional Airy beam. Mathematically, thissymmetric Airy beam (SAB) arises from a finite-energyAiry beam by only changing the odd parity by the evenparity in its spectral cubic phase. We perform a numeri-cal analysis showing its main properties. Besides, theSAB is compared with other kinds of autofocusingAiry-like beams such as DABs and CABs, and possibleapplications are also discussed. Finally, we experimen-tally demonstrate the generation of a symmetric Airybeam.
A key feature of a finite-energy Airy beam is the rela-tionship between its initial profile (ip) in the coordinatespace x, its Gaussian amplitude (Ga), and cubic phase(cp) in the spectral (conjugate) space K . At the initialstage of propagation z � 0, the Fourier transform ofthe spatial field u�x; 0� � u0 is the spectral fieldU�K; 0� � U0 ≡ F fu0g, where F is the Fourier transform
operator. Both conjugate fields are explicitly given by[13]:
u0 � eaxx0Ai�x∕x0�|������{z������}
ip
⇌U0 � x0ea33 e−aK
2x20|�{z�}Ga
eiK3x3
03|{z}
cp
eiKa2x0 ; (1)
where x0 accounts for the size of the Airy central lobe,and the parameter a > 0 is the exponential truncationfactor that guarantees the square integrability of thebeam and controls its spreading properties [1,14]. The lin-ear phase term exp�iKa2x0� has no real effect in thebeam properties for the range of awhere the self-bendingdynamic dominates (a ≪ 1) and in practice, is not con-sidered. The SAB is obtained by only changing the parityin the spectral cubic phase, i.e., K3x30∕3 → jK j3x30∕3, re-sulting in the one-dimensional (1D) and two-dimensional(2D) wrapped phase masks shown in Fig. 1. In this way,the beam intensity and spatial phase are even functionsof the transverse spatial coordinate x. Ananalytic expression for an SAB was not found. However,its paraxial solution v�x; z� can easily be built by usingthe angular spectrum formalism [15]
Dimensionless variables and parameters (~z � z∕λ,~x � x∕λ, ~x0 � x0∕λ, ~K � λK , being λ the wavelength)are used to simplify the numerical integration of Eq. (2),but for a more comprehensive interpretation, the figuresare depicted in terms of the current transverse and longi-tudinal parameters often used in Airy beam literature,s � x∕x0 and ξ � λz∕�2πx20�. Figure 2(a) shows the inten-sity distribution of a 1D SAB, Iv ≡ jvj2, as a function of�s; ξ� and Figs. 2(b)–2(f) show intensity patterns of a2D SAB as a function of transverse spatial variables�sx; sy� for several planes ξ � const. Notice that the beam
energy is mostly concentrated in a single central lobeduring the initial stages of propagation. The maximumintensity, or peak intensity as a function of ξ, i.e.,Imaxv ≡ Imax
v �ξ�, rapidly increases as ξ increases up toreach its extreme value at the autofocus, located atξ � ξf . We name the intensity of the SAB at autofocusby Ifv ≡ Imax
v �ξf �. For the parameters used in the simula-tions ~x0 � 100 and a � 0.08, it has ξf ≈ 1.9 for both 1Dand 2D cases. For ξ > ξf , Imax
v quickly decreases as ξincreases so that the on-axis lobe collapses just afterthe autofocus. At this stage, the beam splits into twospecularly symmetric off-axis lobes whose peak inten-sity follows a parabolic trajectory, which resemblesthose lobes corresponding to two conventional Airybeams accelerating in opposite directions. This is clearlyobserved in Fig. 2(a) for the 1D case. For the 2D case, thecentral lobe becomes, after collapsing, four off-axisidentical lobes with the peak intensity following para-bolic trajectories on the planes x � �y and moving awayfrom the optical axis (z axis) as the beam propagates. InFigs. 2(b)–2(f), the intensity pattern at each plane ξ � ξiwas normalized to the peak intensity on its own plane,i.e., Imax
v �ξi�, and not to the intensity at autofocus, Ifv.This is to help the visualization of the beam pattern struc-ture on each transverse plane. Notice that behind theautofocus, the beam pattern evolves from an internalvertex of peak intensity at ξ � 1.5ξf to an external vertexat ξ � 2ξf in a square symmetry.
On the other hand, and for comparative purposes, westudy the DAB. This beam is a superposition of the com-plex fields of two conventional Airy beams, u� and u
−,
each one accelerating in an opposite direction of x,respectively. Mathematically, the DAB intensity takesthe form Id � ju��s; ξ� � u
−�s; ξ�j2. Figure 3(a) shows
the intensity distribution for a 1D DAB as a functionof �s; ξ�. The intensity is normalized to the peak intensityof an SAB at the autofocus, i.e., Id∕I
fv. Figures 3(b)–3(f)
show intensity patterns for a 2D DAB normalized to themaximum intensity of an SAB at each plane ξ � ξi, i.e.,
Fig. 1. 1D and 2D wrapped spectral phase masks for a conven-tional Airy beam and for the resulting SAB when the phaseparity is changed. The unwrapped phase ranges correspondsto �−20π; 20π�. Fig. 2. (a) Intensity distribution for a 1D symmetric Airy
beam versus �s; ξ�. The intensity scale �0; 1� corresponds to�0; Ifv�. (b)–(f) Intensity patterns for a 2D SAB as a functionof �sx; sy� for several planes ξi � 0, ξf , 1.5ξf , 2ξf , 2.5ξf , ξf beingthe distance between the initial plane and the autofocus plane.The intensity scale �0; 1� for each 2D pattern corresponds to�0; Imax
v �ξi��. All these intensity distributions were obtained bynumerical integration of Eq. (2).
Id�ξi�∕Imaxv �ξi�, as a function of �sx; sy�. The autofocus for
DABs in Fig. 3 is also located at ξ � ξf , and the energycontent of such DABs was set to be equal to SABs ofFig. 2. There are qualitative resemblances between bothbeams. However, there are important quantitativedifferences. At ξf the DAB has 12% less energy concen-trated in the central lobe Its peak intensity is 39% smallerand has a peak 60% wider than that corresponding to theSAB. Moreover, after ξf the DAB spreads in several sec-ondary lobes while the SAB holds two main lobes thatspread according to the central lobes of Airy beamsfor large ξ. Figure 4 depicts the maximum intensity (peakintensity) for symmetric, dual, and conventional Airybeams as a function of ξ. It is clear that SAB and DABhave approximately the same peak intensity at the initialstage of propagation ξ � 0. But at ξ � ξf the peakintensity is higher for SAB. For a 1D case, the ratiois Ifv∕Ifd ≈ 1.5, while this ratio is greater for the 2D caseIfv∕Ifd ≈ 2.7. On the other hand, for large ξ, Imax
v and Imaxu
have identical behavior being both larger than Imaxd .
Thereby, the SAB has better nondiffracting propertiesif compared with the DAB. Furthermore, SABs are aseasily realizable as a conventional Airy beam whileDABs require an extremely complex experimental align-ment task.On the other hand, the only Airy-like beam with
autofocusing properties experimentally demonstratedwas the CAB [9,10], that only exists for two transverse
dimensions because of its circular-cylindrical symmetry.One of the main characteristics of CABs is that the ratioautofocus/initial peak intensity can reach several ordersof magnitude [6–8], while such a ratio is about one orderof magnitude for a 2D SAB, as shown in Fig. 4. However,this should be not seen as a disadvantage for SABs sincethe propagation dynamics of both classes of beams isdifferent. The CAB is initially formed as a successionof Airy rings being the inner ring, the principal one. Thisring bends as propagates to a focal point on the axis ofsymmetry. After the autofocus, the energy is mostlyconcentrated in a central on-axis lobe as z increases(see for instance Fig. 2 in [7]). This process is dynami-cally inverse to what happens with SABs because, in thislast case, the autofocus is generated from the central lobewhile in the CAB case, the central lobe is a result of thebeam collapse. This inverse propagation behavior ofSABs with respect to CABs, in addition to their differentspatial symmetries, lead to us to think that both beamsmay be seen as complementary for practical applications.SABs could also be useful in optical micromanipulationrequiring symmetric Airy patterns as pointed out in [12].In curved channels of plasma [16], the SAB seems to bepromising since it presents bifurcation and high intensitycontrast (autofocusing) without any nonlinear effect.Besides, an advantage in relation to CABs is that SABscan also exist in 1D as planar beams opening the chanceof developing applications to plasmonics [17].
Finally, we demonstrate the experimental realizationof an SAB. A great advantage is that it can be generatedsimilarly to a conventional Airy beam. The beam’scomplex field amplitude V0 in Eq. (3) was encoded asa phase-only computer generated hologram (CGH)following the approach reported in [18]. Note that, alter-natively to the CGH, the SAB can also be generated bymodulating a Gaussian beam with the correspondingeven cubic phase function. In our case, the CGH wasaddressed into a programmable reflective LCoS-SLM(Holoeye PLUTO, 8-bit gray-level, pixel pitch of 8 μmand 1920 × 1080 pixels) calibrated for a 2π phase shift atthe wavelength λ � 532 nm and corrected from staticaberrations as reported in [19]. To generate the SAB,the hologram was illuminated by a collimated laser beam(λ � 532 nm) and then focused by a spherical convergentlens (focal length of 10 cm, N-BK7 glass). The beampropagation in the focal region of such a focusinglens has been measured by using an sCMOS camera(Hamamatsu, Orca Flash 4.0, 16-bit gray-level, pixel sizeof 6.5 μm) and stored as a video (Media 1). Specifically,we measured 240 intensity patterns in the rangez ∈ �0; 20� mm, where z � 0 coincides with the focalplane position of the focusing lens. We use dimensionalvariables to illustrate the real propagation range andbeam-size values of the generated SAB. In Fig. 5(a) weshow the beam profile in the x–z axes where the auto-focus region and the off-axis lobes are clearly distin-guished. The intensity patterns at the distance zi � 0,zf , 1.5zf , 2zf , 2.5zf are displayed in Figs. 5(b)–5(f) beingzf � 5.7 mm, the distance between the initial and theautofocus planes. Notice that these experimentalpatterns are in good agreement with the theoretical onesdisplayed in Fig. 2. These results experimentally confirmthe main features of an SAB.
Fig. 3. (a) Intensity distribution for a 1D dual Airy beam(DAB) versus �s; ξ�. The intensity scale �0; 1� corresponds to�0; Ifv�. (b)–(f) Intensity patterns for a 2D DAB versus �sx; sy�at the planes ξi � 0, ξf , 1.5ξf , 2ξf , 2.5ξf . The intensity scale�0; 1� corresponds to �0; Imax
v �ξi��. All these intensity distributionswere obtained from the superposition of well-known analyticalexpressions for the Airy beam with opposite accelerations.
Fig. 4. Normalized peak intensity for SAB, Imaxv ∕Ifv, DAB
In summary, a new class of Airy beams having rectan-gular symmetry and exhibiting autofocusing features waspresented. The SAB arises from the symmetrization ofthe spectral cubic phase of a conventional Airy beam.The SAB can be generated in 1D and 2D as well. Its prop-erties were analyzed and compared to those correspond-ing to other kinds of Airy-like autofocusing beams, DABsand CABs, and potential applications were discussed.The symmetric Airy beam was experimentally demon-strated in good agreement with the theoretical predic-tions. This new type of beam opens the possibility ofstudying different ways of trapping and guiding particlesor using them to generate new structured plasmachannels and waveguides in dielectric media as wellas in plasmonic applications.
Financial support from the SENAI-DR/Bahia, Brazil,and the Spanish MEC under project TEC 2011-23629 isacknowledged. P. V. acknowledges a PQ fellowship fromCNPq (Brazil).
References and Note
1. G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979(2007).
2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N.Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
3. Y. Kaganovsky and E. Heyman, Opt. Express 18, 8440(2010).
4. Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N.Christodoulides, and Z. Chen, in Nonlinear Photonicsand Novel Optical Phenomena, Z. Chen and R. Morandotti,eds., Vol. 170 of Springer Series in Optical Sciences(Springer, 2013), pp. 1–46.
5. M. A. Bandres, I. Kaminer, M. S. Mills, B. M. Rodrguez-Lara,E. Greenfield, M. Segev, and D. N. Christodoulides, Opt.Phot. News 24(6), 30 (2013).
6. N. K. Efremidis and D. N. Christodoulides, Opt. Lett. 35,4045 (2010).
7. I. D. Chremmos, N. K. Efremidis, and D. N. Christodoulides,Opt. Lett. 36, 1890 (2011).
8. I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K.Efremidis, Phys. Rev. A 85, 023828 (2012).
9. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides,and S. Tzortzakis, Opt. Lett. 36, 1842 (2011).
10. I. D. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N.Christodoulides, and Z. Chen, Opt. Lett. 36, 3675 (2011).
11. P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S.Tzortzakis, Nat. Commun. 4, 2622 (2013).
12. Ch.-Y. Hwang, D. Choi, K.-Y. Kim, and B. Lee, Opt. Express18, 23504 (2010).
13. All the equations are given for one transverse dimension.However, the extension to two transverse dimensions isstraightforward because of the rectangular symmetry.
14. I. M. Besieris and A. M. Shaarawi, Opt. Lett. 32, 2447(2007).
15. P. Vaveliuk and O. Martinez Matos, Opt. Express 20, 26913(2012).
16. J. Kasparian and J.-P. Wolf, J. Eur. Opt. Soc. (Rapid Publi-cations) 4, 09039 (2009).
17. A. Salandrino and D. N. Christodoulides, Physics 4, 69(2011).
18. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I.Moreno, Appl. Opt. 38, 5004 (1999).
19. J. A. Rodrigo, T. Alieva, A. Cámara, O. Martínez-Matos, P.Cheben, and M. L. Calvo, Opt. Express 19, 6064 (2011).
Fig. 5. (a) Experimental intensity profile on the xz plane for a2D symmetric Airy beam versus �x; z�. The intensity scale �0; 1�corresponds to �0; Ifv�. (b)–(f) Experimental intensity patternsfor a 2D SAB as a function of �x; y� for different planes zi.The density scale �0; 1� for each 2D pattern corresponds to�0; Imax
v �zi��. The SAB propagation from z � 0 to 20 mm isshown in Media 1.
